Proportion and similarity question

Discussion in 'Community Discussion' started by Jasonbot, Nov 28, 2007.

  1. Jasonbot macrumors 68020

    Jasonbot

    Joined:
    Aug 15, 2006
    Location:
    The Rainbow Nation RSA
    #1
    Hi everyone. COuld you guys please help me with this simple geometry problem?

    In triangleADC, angleD=90*, DB perpendicular AC, BK||AD and 2DK=KC
    Using triangleDAB ||| triangleCAD (already proven using AAA) prove that 2DA=sqrt3.BC

    thanks,
    Jason
     

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  2. Jasonbot thread starter macrumors 68020

    Jasonbot

    Joined:
    Aug 15, 2006
    Location:
    The Rainbow Nation RSA
    #2
    Don't they teach geometry in foreign countries?
     
  3. swiftaw macrumors 603

    swiftaw

    Joined:
    Jan 31, 2005
    Location:
    Omaha, NE, USA
    #3
    OK,

    AB^2 + BD^2 = AD^2
    BC^2 + BD^2 = CD^2

    Putting these together we get that

    AB^2 + CD^2 - BC^2 = AD^2 (*)

    Now, by similarity we have that AC = 3AB, so AC^2 = 9AB^2.

    Using that fact (*) becomes
    (1/9)AC^2 + CD^2 - BC^2 = AD^2

    But AC^2 = AD^2 + CD^2, thus

    (1/9)(AD^2 + CD^2) + CD^2 - BC^2 = AD^2

    Re-arranging we get
    (10/9)CD^2 - BC^2 = (8/9)AD^2

    Now, use the fact that CD^2 = AC^2 - AD^2 we get

    (10/9)(AC^2 - AD^2) - BC^2 = (8/9)AD^2
    or, in other words
    2AD^2 = (10/9)AC^2 - BC^2 (**)

    But, AC = (3/2)BC, so substituting into (**) gets us

    2AD^2 = (10/9)(9/4)BC^2 - BC^2 = (6/4)BC^2

    Doubling both sides

    4AD^2 = (12/4)BC^2 = 3BC^2

    Taking square roots

    2AD = sqrt(3)*BC
     
  4. Jasonbot thread starter macrumors 68020

    Jasonbot

    Joined:
    Aug 15, 2006
    Location:
    The Rainbow Nation RSA
    #4
    Thank you so much! I have no idea how you see these things but I'm sure Ill get it :confused: DO you look at what you're supposed to prove and work backwards?
     
  5. swiftaw macrumors 603

    swiftaw

    Joined:
    Jan 31, 2005
    Location:
    Omaha, NE, USA
    #5
    No, that is bad form. The only thing I looked in the final answer was that it only contained AD and BC. I started by writing down things I knew to be true and then tried to combine them in such a way to leave only AD and BC.
     

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